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Let's note that standard number 0 also appears, according to this definition, infinitesimal. But all other infinitesimal numbers cannot the standard. It follows from this that for standard numbers Archimedes's axiom is fair.

More exact definition of an infinite trifle of number> 0 which we will use vdalneyshy such is. Let's put number with ourselves, receiving numbers + etc. If all received numbers appear less than 1, the number and will be called infinitesimal. In other words, if infinitely it is not enough, how many times do not postpone length piece along a piece of length 1, up to the end you will not reach. Our requirement to infinitesimal can be copied in such form

So, it will be a question of infinitesimal numbers. What number it is necessary to call infinitesimal? Let's assume that this positive number, if it most less positive numbers. It is easy to understand that such does not happen: if it is more than zero, it is one of positive numbers therefore our definition demands that the number was less than itself. Therefore we will demand that was the smallest in a set of positive numbers. On a numerical axis such has to be represented by the most left point of a set. Unfortunately the number with the specified properties too is not present and cannot be: the number will be positive number, smaller.

Is as follows: if we want to consider a conclusion infinitesimal, we have to expand a set of R real numbers to some big set * with R. We will call elements of this new set hyper real numbers. In it Archimedes's axiom is not carried out and there are infinitesimal numbers, such that how many them do not put with yourself, the sum will remain less Non-standard all the time, or not Archimedean, the analysis studies a set of hyper real numbers * R.

Let there is some set P, in it some elements 0 and 1 are allocated and the operations of addition, subtraction, multiplication and division putting in compliance to two any elements and sets P their sum, work, a difference and private (if) are defined. Let thus the listed operations possess all usual properties.

The non-standard analysis would remain a curious funny thing if justification of reasonings of classics of the mathematical analysis was its only appendix. It was useful and at development of new mathematical theories. The non-standard analysis can be compared to the bridge thrown through the river. Construction of the bridge does not broaden the territory available to us, but reduces a way from one coast by another. In this way the non-standard analysis does proofs of many theorems shorter.

Let's assume that required expansion * is already constructed by R, and we investigate its structure. * R we will call set elements hyper real numbers. Among them also all real numbers contain. To distinguish them, we will call real numbers (elements R) standard, and other hyper real numbers (elements * R/R) — non-standard.

First of all, we receive not Archimedean expansion of a field of real numbers. Besides, its analog in "the non-standard world" is delivered to "each object of the standard world" in compliance. Non-standard analog of any real number is it; to any subset And sets of R there corresponds the subset * And sets * to R, each function f from R in R there corresponds function * to f from * R in * R, each two-place function g from R in R there corresponds function * to g from * R in * R etc. Certainly, these analogs * A, * f, * are not any g, and have to possess some special properties: so, * And, on real numbers of f and * f coincide so * f is continuation for f, and * g - continuation for g. Thus it is executed the so-called principle of transfer claiming roughly saying that the hyper valid analogs of standard objects possess the same properties, as initial standard objects.

In that case the set P is called as a field. Let in the field P the order be entered, i.e. for any couple of elements not equal each other and is defined which of them is more. Thus such properties are carried out:

. That over hyper real numbers it was possible to carry out usual operations: any two hyper real numbers need to be able to be put, multiplied, read and divided and so that usual properties of addition and multiplication were carried out. Besides, it is necessary to be able to compare hyper real numbers in size, i.e. to solve what of them more.